Majorization revisited: Comparison of norms in interpolation scales

We reformulate, modify and extend a comparison criteria of L norms obtained by Nazarov-Podkorytov and place it in the general setting of interpolation theory and majorization theory. In particular, we give norm comparison criteria for general scales of interpolation spaces, including noncommutative L and Lorentz spaces. As an application, we extend the classical Ball’s integral inequality, which lies at the basis of his famous result on sections of the n−dimensional unit cube.

[1]  Jöran Bergh,et al.  Interpolation Spaces: An Introduction , 2011 .

[2]  A. Khintchine.  Über dyadische Brüche , 1923 .

[3]  M. Ledoux,et al.  Logarithmic Sobolev Inequalities , 2014 .

[4]  J. Pečarić,et al.  On an integral inequality. , 2004 .

[5]  C. Bennett,et al.  Interpolation of operators , 1987 .

[6]  L. Persson,et al.  Convexity, concavity, type and cotype of Lorentz spaces ☆ , 1998 .

[7]  B. Jawerth,et al.  The domain spaces of quasilogarithmic operators , 1990 .

[8]  An interpolation theorem for modular spaces , 1984 .

[9]  Joram Lindenstrauss.  Classical Banach Spaces II: Function Spaces , 1979 .

[10]  J. Peetre,et al.  Interpolation and non-commutative integration , 1975 .

[11]  Yu. L. Ershov,et al.  The theory of A-spaces , 1973 .

[12]  J. Peetre,et al.  Interpolation of normed abelian groups , 1972 .

[13]  Joram Lindenstrauss,et al.  Classical Banach spaces , 1973 .

[14]  M. Milman,et al.  Extrapolation: Stories and Problems , 2020, 2004.03655.

[15]  K. Ball.  Some remarks on the geometry of convex sets , 1988 .

[16]  F. Nazarov,et al.  Ball, Haagerup, and Distribution Functions , 2000 .

[17]  R. Culverhouse,et al.  Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions , 2002 .

[18]  Elias M. Stein,et al.  Interpolation of linear operators , 1956 .

[19]  M. Kreĭn,et al.  Introduction to the theory of linear nonselfadjoint operators , 1969 .

[20]  Sergey V. Astashkin,et al.  Limiting interpolation spaces via extrapolation , 2018, J. Approx. Theory.

[21]  L. Asimow.  Interpolation in Banach spaces , 1979 .

[22]  L. Persson,et al.  Type, cotype and convexity properties of Orlicz spaces , 1997 .

[23]  J. Wissel,et al.  On the Best Constants in the Khintchine Inequality , 2007 .

[24]  H. König.  On the best constants in the Khintchine inequality for Steinhaus variables , 2014 .

[25]  A. Calderón.  Spaces between $L^{1}$ and $L^{∞}$ and the theorem of Marcinkiewicz , 1966 .

[26]  S. Mazur.  Über schwache Konvergenz in den Raümen ($L^{p}$) , 1933 .

[27]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[28]  Olaf Mordhorst.  The optimal constants in Khintchine's inequality for the case 2 , 2016, 1601.07850.

[29]  L. Maligranda,et al.  Order convexity and concavity of Lorentz spaces $Λ_{p,w}$, 0 < p < ∞ , 2004 .

[30]  T. Tkocz,et al.  From Ball's cube slicing inequality to Khinchin-type inequalities for negative moments , 2020, 2011.12251.

[31]  Alexandros Eskenazis,et al.  Sharp comparison of moments and the log-concave moment problem , 2018, Advances in Mathematics.

[32]  Philip D. Plowright,et al.  Convexity , 2019, Optimization for Chemical and Biochemical Engineering.

[33]  R. Jackson.  Inequalities , 2007, Algebra for Parents.

[34]  G. Lorentz,et al.  INTERPOLATION THEOREMS FOR THE PAIRS OF SPACES (LP,L°°) AND (L1,L9)(1) , 2010 .